weighted geometric mean

{{Short description|Generalization in statistics mathematics}}

In statistics, the weighted geometric mean is a generalization of the geometric mean using the weighted arithmetic mean.

Given a sample $x=(x\_1,x\_2\backslash dots,x\_n)$ and weights $w=(w\_1,\; w\_2,\backslash dots,w\_n)$, it is calculated as:{{citation

| last = Siegel | first = Irving H.

| date = June 1942

| doi = 10.1080/01621459.1942.10500636

| issue = 218

| journal = Journal of the American Statistical Association

| pages = 271–274

| title = Index-number differences: geometric means

| volume = 37}}

:$\backslash bar\{x\}\; =\; \backslash left(\backslash prod\_\{i=1\}^n\; x\_i^\{w\_i\}\backslash right)^\{1\; /\; \backslash sum\_\{i=1\}^n\; w\_i\}\; =\; \backslash quad\; \backslash exp\; \backslash left(\; \backslash frac\{\backslash sum\_\{i=1\}^n\; w\_i\; \backslash ln\; x\_i\}\{\backslash sum\_\{i=1\}^n\; w\_i\; \backslash quad\}\; \backslash right)$

The second form above illustrates that the logarithm of the geometric mean is the weighted arithmetic mean of the logarithms of the individual values.

If all the weights are equal, the weighted geometric mean simplifies to the ordinary unweighted geometric mean.